document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189589>Question 1189589</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A particle moves in a straight line with velocity V(t)=root(3t-1) meters per second where t is time in seconds. At t=2, the particle's distance from the starding point was 8 meters in the positive direction. What is the particle's position at t=7 seconds? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849270 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849270\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem:\r<BR>\n" );
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document.write( "1. **Find the displacement function:**\r<BR>\n" );
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document.write( "The velocity function V(t) is the derivative of the position function s(t). To find the position function, we need to integrate the velocity function:\r<BR>\n" );
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document.write( "s(t) = ∫V(t) dt = ∫√(3t - 1) dt\r<BR>\n" );
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document.write( "Let u = 3t - 1, so du = 3 dt, and dt = du/3.  Substituting:\r<BR>\n" );
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document.write( "s(t) = ∫√u * (du/3) = (1/3) ∫u^(1/2) du<BR>\n" );
document.write( "s(t) = (1/3) * (2/3) * u^(3/2) + C = (2/9) * (3t - 1)^(3/2) + C\r<BR>\n" );
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document.write( "where C is the constant of integration.\r<BR>\n" );
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document.write( "2. **Solve for the constant of integration (C):**\r<BR>\n" );
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document.write( "We are given that at t = 2, the particle's position is 8 meters.  Plug in these values to solve for C:\r<BR>\n" );
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document.write( "8 = (2/9) * (3*2 - 1)^(3/2) + C<BR>\n" );
document.write( "8 = (2/9) * 5^(3/2) + C<BR>\n" );
document.write( "8 = (2/9) * 5√5 + C<BR>\n" );
document.write( "C = 8 - (10√5)/9\r<BR>\n" );
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document.write( "3. **Find the position at t = 7:**\r<BR>\n" );
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document.write( "Now that we have the complete position function, we can find the position at t = 7:\r<BR>\n" );
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document.write( "s(7) = (2/9) * (3*7 - 1)^(3/2) + 8 - (10√5)/9<BR>\n" );
document.write( "s(7) = (2/9) * 20^(3/2) + 8 - (10√5)/9<BR>\n" );
document.write( "s(7) = (2/9) * 20√20 + 8 - (10√5)/9<BR>\n" );
document.write( "s(7) = (40√5)/9 + 8 - (10√5)/9<BR>\n" );
document.write( "s(7) = (30√5)/9 + 8<BR>\n" );
document.write( "s(7) = (10√5)/3 + 8<BR>\n" );
document.write( "s(7) ≈ (10 * 2.236) / 3 + 8<BR>\n" );
document.write( "s(7) ≈ 7.453 + 8<BR>\n" );
document.write( "s(7) ≈ 15.453\r<BR>\n" );
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document.write( "Therefore, the particle's position at t = 7 seconds is approximately 15.45 meters.<BR>\n" );
document.write( " </SPAN>\n" );
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